# Which square numbers make 96

## Quadratic text equations - number puzzles

Quadratic text equations - number puzzles - solutions 1. If you add its square number to a number, you get 56. Number: x x + x & sup2; = 56 has the solution: x1 = –8 x2 = 7 Answer: The number is 8 or –7. 2. If you add its square number to a number, you get 132. Number you are looking for: x x + x & sup2; = 132 has the solution: x1 = 11 x2 = –12 Answer: The number is 11 or –12. 3. If you multiply two consecutive numbers, you get 812. What are the names of the numbers? 1st number: x 2nd number: x + 1 x (x + 1) = 812 has the solution: x1 = 28 x2 = –29 The numbers are 28 and 29 or –29 and –28. 4. The product of two consecutive numbers is 552. 1st number: x 2nd number: x + 1 x (x + 1) = 552 has the solution: x1 = 23 x2 = –24 The numbers we are looking for are 23 and 24 or -24 and -23. 5. If you multiply a number by the number larger by 5, you get 374. 1st number: x 2nd number: x + 5 x (x + 5) = 374 has the solution: x1 = 17 x2 = –22 The numbers we are looking for are 17 and 22 or –22 and –17. 6. Of two numbers, one is 7 smaller than the other. Your product is 368. 1st number: x 2nd number: x - 7 x (x - 7) = 368 has the solution: x1 = 23 x2 = –16 The numbers you are looking for are 23 and 16 or - 16 and -23. 7. The sum of two numbers is 15, their product is 56. 1st number: x 2nd number: 15 - xx (15 - x) = 56 has the solution: x1 = 7 x2 = 8 The numbers we are looking for are 7 and 8 (or 8 and 7). 8. The sum of two numbers is 33, their product is 162. 1st number: x 2nd number: 33 - xx (33 - x) = 162 has the solution: x1 = 6 x2 = 27 The numbers we are looking for are 6 and 27 (or 27 and 6). 9. The difference between two numbers is 4, their product is 221. 1st number: x 2nd number: x + 4 x (x + 4) = 221 has the solution: x1 = 13 x2 = –17 The numbers we are looking for are 13 and 17 and -17 and -13, respectively. 10. The squares of two numbers add up to 164. The two numbers differ by 2. 1st number: x 2nd number: x + 2 x & sup2; + (x + 2)? = 164 has the solution: x1 = 8 x2 = –10 The numbers we are looking for are 8 and 10 or –10 and –8. 11. Two numbers differ by 5. The sum of their squares is 625. 1st number: x 2nd number: x + 5 x & sup2; + (x + 5)? = 625 has the solution: x1 = 15 x2 = –20 The numbers we are looking for are 15 and 20 or –20 and –15. 12. If you increase each factor by the same number in the product of 13 and 17, you get 396 as the result. The number you are looking for: x (13 + x) (17 + x) = 396 has as Solution: x1 = 5 x2 = -35 The factors were increased by 5 or -35. 13. If you reduce each factor by the same number in the product of 23 and 35, the result is 589. The number you are looking for: x (23 - x) (35 - x) = 589 has the solution: x1 = 4 x2 = 54 The factors were reduced by 4 and 54 respectively. 14. The product of two numbers is 1596. One number is just as far above 40 as the other below 40. Number: x (40 + x) (40 - x) = 1596 has the solution: x1 = 2 x2 = –2 The number 40 must be increased or decreased by 2 (or –2). Thus the two numbers are called 42 and 38 (or 38 and 42). 15. The product of two numbers is 899. One number is just as far above 30 as the other below 30. The number you are looking for: x (30 + x) (30 - x) = 899 has the solution: x1 = 1 x2 = –1 The number 30 must be increased or decreased by 1 (or -1). Thus the two numbers are called 31 and 29 (or 29 and 31). 16. If you add the reciprocal values of two consecutive natural numbers, you get 7. 12 1st number: x 2nd number: x + 1 1 1 7 xx 1 12 has the solution: x1 = 3 4 x2 = (not applicable) 7 The numbers you are looking for are 3 and 4 17. Two natural numbers differ by 2. If you add their reciprocal values, the result is 12. 35 1st number: x 2nd number: x + 2 1 1 12 xx 2 35 has as solution: x1 = 5 7 x2 = - not applicable as solution 6 The numbers are 5 and 7 18. Adding 96 to the square of a number gives 321. Number: x x & sup2; + 96 = 321 has the solution: x1 = 15 x2 0 = -15 The number is 15 or -15. 19. The square of the sum of a number and the number 5 has the value 64. Number you are looking for: x (x + 5) & sup2; = 64 has the solution: x1 = 3 x2 = –13 20. Split 96 into two integer factors, the difference of which is 4. 1st number: x 2nd number: x - 4 x (x 4) 96 has the solution: x1 = –8 x2 = 12 The numbers are –8 and –12 or 8 and 12. 21. For which numbers a does the equation x & sup2; + 22x + a = 0 a) no solution, b) one solution, c) two solutions? From 22 484 4 a x1; 2 2 we get: a & gt; 121: no solution a = 121: a solution a & lt; 121: two solutions 22. What are the names of the numbers whose square is 24 times larger than their 10-fold? Number sought: x x & sup2; - 24x = 10x has the solution: x1 = –2 x2 = 12 The number is –2 or 12. 23. The sum of two numbers is 50. If you decrease the first number by 4, you increase the second Number by 4, this creates two new numbers. If you now form the reciprocal values of these new numbers, then the sum of the 1 reciprocal values is. 12 What are the original numbers? 1st number: x 2nd number: y I. x y 50 1 1 1 x 4 54 x 12 leads to: x & sup2; - 58x + 816 = 0 has as the solution: x1 = 24 (y1 = 26) x2 = 34 (y2 = 16) The numbers are 24 and 26 or 34 and 16. II. 24. a) One increases a number around its square number, you get 30. What is the name of the number? b) b) Which number is 42 smaller than its square number? a) number sought: x b) number sought: x x & sup2; x 30 x & sup2; 42 x x & sup2; x 30 0 x & sup2; x 42 0 a 1; b 1; c 30 a 1; b 1; c 42 1 1 20 2 1 11 x1.2 2 x1 5; x 2 6 x1.2 25. 1 1 168 2 1 13 x1.2 2 x1 7; x 2 6 x1,2 a) Which number has to be multiplied by the number larger by 4 to get the product 21? b) b) Which number has to be multiplied by the number smaller by 5 to get the product 36? a) Number you are looking for: x b) Number you are looking for: x x x 4 21 x x 5 36 x & sup2; 4x 21 0 p 2; q 21 2 x1.2 2 4 21 x1.2 2 5 x1 3; x 2 7 The numbers are 3 and 7 or –3 and –7. x? 5x 36 0 a 1; b 5; c 36 5 25 144 2 5 13 x1.2 2 x1 9; x 2 4 x1.2 The numbers are 9 and 4 or –4 and –9. 26. Which two consecutive integers have the product 56 (132)? 1st number: x 2nd number: x + 1 x x 1 56 1st number: x 2nd number: x + 1 x x 1 132 x & sup2; x 56 0 a 1; b 1; c 56 x & sup2; x 132 0 a 1; b 1; c 132 1 1 224 2 1 15 x1.2 2 x1 7; x 2 8 x1.2 The numbers are 7 and 8 or –7 and –8. 1 1 528 2 1 23 x1.2 2 x1 11; x1,2 x 2 12 The numbers are 11 and 12 or –11 and –12. 27. a) Break 48 down into two factors, the sum of which is 14. b) Split 48 into two factors, the difference of which is 8 1st factor: x 1st factor: x 2nd factor: 14 - x 2nd factor: x - 8 x 14 x 48 x x 8 48 x & sup2; 14x 48 0 p 7; q 48 2 x & sup2; 8x 48 0 p 4; q 48 2 x1.2 7 49 48 x1.2 4 16 48 x1.2 7 1 x1.2 4 8 x1 8; x 2 6 x1 12; The factors are 8 and 6. The factors are 12 and 4 and –12 and –4, respectively. x 2 4 28. Split 8 (12) into two summands, the squares of which add up to 40 (90). 1st summand: x 2nd summand: 8 - x x & sup2; 8 x & sup2; 40 1st summand: x 2nd summand: 12 - x x & sup2; 12 x & sup2; 90 x & sup2; 8x 12 0 p 4; q 12 2 x & sup2; 12x 27 0 p 6; q 27 2 x1.2 4 16 12 x1.2 6 36 27 x1.2 4 2 x1.2 6 3 x1 6; x 2 2 x1 9; x 2 3 The summands are 6 and 2. The summands are 9 and 3. 29. If you reduce each factor in the product 11 9 (17 23) by a number, the product becomes 35 (160). What is the name of the number? Number sought: x Number sought: x 11 x 9 x 35 17 x 23 x 160 x & sup2; 20x 64 0 p 10; q 64 2 x & sup2; 40x 231 0 p 20; q 231 2 x1.2 10 100 64 x1.2 20 400 231 x1.2 10 6 x1.2 20 13 x1 16; x2 4 The number is 16 or 4. x1 33; x2 7 The number is 33 or 7. 30. If you add its inverse number to a number, you get 5 29 . What is the name of the 2 10 number? Wanted number: x 1 5 x x 2 2x & sup2; 5x 2 0 Number you are looking for: x 1 29 x x 10 10x & sup2; 29x 10 0 a 10; b 29; c 10 a 2; b 5; c 2 5 25 16 4 53 x1,2 4 1 x1 2; x 2 2 29 841 400 20 29 21 x1.2 20 5 2 x1 ; x 2 2 5 x1.2 The numbers are 2 and x1.2 1. 2 The numbers are 5 2 and. 2 5 31. The numerator of a fraction is 3 (2) smaller than the denominator. Adding to 5 34 . What is the name of the fraction? 2 15 counter: x x x 2 34 x2 x 15 x & sup2; 2x 15 0 the fraction has its reciprocal value, one obtains counter: x x x3 5 x3 x 2 x & sup2; 3x 18 0 a 1; b 3; c 18 3 9 72 2 3 9 x1.2 2 x1 3; x 2 6 x1.2 3 6. The fraction is resp. 6 3 p 1; q 15 2 x1.2 1 1 15 x1.2 1 4 x1 3; x 2 5 The fraction is 3 5 respectively. 5 3 32. The sum of the squares of two numbers is 3.2; their arithmetic mean 1.2. What are the numbers? 1st number: x 2nd number: y x & sup2; y & sup2; 3.2 xy 1.2 2 2.4 y 2 y & sup2; 3.2 y & sup2; 2.4y 1.28 0 y1 1.6; y 2 0.8 x1 0.8; x 2 1.6 The numbers are 1.6 and 0.8 and 0.8 and 1.6, respectively. 33. Two numbers behave like 5: 7. The sum of their squares is 1,850. What are the numbers? 1st number: x 2nd number: y x: y 5: 7 x & sup2; y & sup2; 1850 5 7 y & sup2; y & sup2; 1850 y & sup2; 1225 0 y1 35; y 2 35 x1 25; x 2 25 The numbers are 35 and 25 or –35 and –-25. 34. The product of two numbers is 836. One number is just as much above 30 as the other is below 30. What are the names of the numbers? Searched number: x 30 x 30 x 836 x & sup2; 64 0x1 8; x 2 8 Numbers you are looking for: 38 and 22 or 22 and 38. 35. The third and fourth parts of a number are squared and then added together 12.25. What is the name of the number? Searched number: x x x 3 & sup2; 4 & sup2; 12.25 x & sup2; 70.56 0x1 8.4; x 2 8.4 The number is 8.4 or –8.4. 36. The sum of the squares of four consecutive even natural numbers is 1176. What are these numbers called? 1st number: x 2nd number: x + 2 3rd number: x + 4 4th number: x + 6 x & sup2; + (x + 2)? + (x + 4)? + (x + 6)? = 1176 x1 = 14 (x2 = –20) The numbers are called 14, 16, 18 and 20. 37. The sum of two numbers is 94, their product is 2109. What are the names of the numbers? 1st number: x 2nd number: y I. x y 94 II. X y 2109 x 57 y 37 The numbers are called 57 and 37. 38. Of two numbers, one is larger by 12 ; & rsquo; he than the other. The product of the two numbers is 864. Calculate the numbers. 1st number: x 2nd number: x + 12 x (x + 12) = 864 x1 = 24 x2 = -36 The numbers are called 24 and 36 or -36 and -24. 39. Of two numbers, one is just as much larger than the other smaller than 35. The product of the two numbers is 1104. What are the names of the numbers? 1st number: 35 + x 2nd number: 35 - x (35 + x) (35 - x) = 1104 x1 = 11 x2 = –11 The numbers are called 46 and 24 or 24 and 46. 40. Um what number must the first factor of the product 16 · 17 should be reduced and the second factor increased so that the value of the product is reduced by 30? Searched number: x (16 - x) (17 + x) = 242 x1 = 5 x2 = –6 The number is 5 or –6. 41. The numerator of a fraction is 3 smaller than the denominator. If you increase the numerator and denominator by 10 at the same time, you get a fraction that is twice as large. is like the first break. What is the name of the bridge? (x 3) 10 x 3 2 x 10 x x1 5 x 2 12 The bridges are called: 2 12 and 5 9

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